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discuss the n-valued function y which is defined, as a function
of #, by the equation
P»Wyn 4- PitoiT1 -f •• • + PR-i(*)ir + P»<«) = 0,
in which the P's are polynomials in %. This equation defines y as
an n-valued function of x. As in the case of z/2 — as = 0, there
will be certain values of x for which two or more of these n
values of y are equal. These values of x are the so-called branch
points of the n-valued function defined by the equation.
All this is now extended to functions of complex variables,
and the function w (also its integral) as defined by
P0(s)»" -r P^)-"-1 -r .. - + P»-i(=)w + PB(c) = 0,
in which z denotes the complex variable s -f it, where s91 are
real variables and i = V — 1. The n values of w are called the
branches of the function aj. Here we must refer (chapter on
Gauss) to what was said about the representation of uniform
functions of z. Let the variable z (= s -j- if) trace out any path
in its plane, and let the uniform function/(z) be expressed in
the form 17 -f iV, where U, V are functions of s, t. Then, to
every value of z will correspond one, and only one, value for
each of U,V, and, as z traces out its path in the ss i-plane, / (z)
will trace out a corresponding path in the Z7, F-plane: the path
of / (2) will be uniquely determined by that of 2. But if w is a
multiform (many-valued) function of z> such that precisely n
distinct values of w are determined by each value of z (except
at branch points, where several values of w may be equal), then
it is obvious that one u^plane no longer suffices (if n is greater
than 1) to represent the path, the 'march' of the function 10. In
the case of a too-valued function to, such as that determined by
a;2 = z> two zt'-planes would be required and, quite generally, for
an n-valued function (n finite or infinite), precisely n such zo-
planes would be required.
The advantages of considering uniform (one-valued) func-
tions instead of n-valued functions (n greater than 1) should be
obvious even to a non-mathematician. What Riemann did was
this: instead of the «»distinct re-planes, he introduced an n-
sheeted surface, of the sort roughly described in what follows,